Explicit solutions of nonlinear evolution equations via nonlocal reductions approach (Q2711766)

The authors present an approach of studying nonlinear PDEs possessing a so-called Zakharov-Shabat representation (generalized Lax representation): NEWLINE\[NEWLINE\beta(L_1)_t-\alpha(L_2)_y+[L_1,L_2]=0,\;L_1=\sum_{i=0}^{p}u_i (t,x,y)\partial^i/\partial x^i, L_2=\sum_{i=0}^{p}v_i(t,x,y)\partial^i/\partial x^i,NEWLINE\]NEWLINE where \(u_i, v_i\) are functions to be found. The main idea is to impose certain additional conditions (reductions) on functions \(u_i, v_i\) in the form of a system of PDE and to reduce the (1+2)-dimensional problem to that of dimension (1+1). This approach is applied to the systemNEWLINE\[NEWLINE[V_t,A]-[V_y,B]+AV_xB-BV_xA+ [[V,A],[V,B]]=0NEWLINE\]NEWLINE which is of great interest for theoretical physics. Here \(A=\text{diag}(a_1,\ldots,a_n)\), \(B=\text{diag}(b_1,\ldots,b_n),\) \(a_i,b_j\) are given real numbers, and \(V(t,x,y)\) is \((n\times n)\)-matrix-function with unknown elements. The authors find an exact solution of the form NEWLINE\[NEWLINEV=\varphi \left (C+\int_{x}^{+\infty }\bar \varphi^T\varphi dx\right)^{-1}\bar \varphi^ T,NEWLINE\]NEWLINE where elements of the matrix \(\varphi \) are represented in the form \(\varphi _{km}=f_{km}(x+a_ky+b_kt)\) with arbitrary functions \(f_{km}(\tau)\) and \(C\) is a constant nondegenerate \((n\times n)\)-matrix satisfying the equality \(\bar C^T=C\).NEWLINENEWLINEFor the entire collection see [Zbl 0937.00046].